Continuum mechanics/Volume change and area change

Volume change edit

Consider an infinitesimal volume element in the reference configuration. If a right-handed orthonormal basis in the reference configuration is   then the vectors representing the edges of the element are


The volume of the element is given by


Upon deformation, these edges go to   where




Therefore, the deformed volume is given by






Recall that from conservation of mass we have


Therefore, an alternative form of the conservation of mass is


Distortional component of the deformation gradient edit

For many materials it is convenient to decompose the deformation gradient in a volumetric part and a distortional part. This is particularly useful when there is no volume change in the material when it deforms - for example in muscles, rubber tires, metal plasticity, etc.

Let us assume that the deformation gradient can be decomposed into a volumetric part and a distortional part, i.e,




Since there is no volume change due to the pure distortion, we have


If   we have


Therefore the distortional component of the deformation gradient is given by


We can use this result to find the distortional components of various strain and deformation tensors. For example, the right Cauchy-Green deformation tensor is given by


If we define its distortional component as


we have


Area change - Nanson's formula edit

Nanson's formula is an important relation that can be used to go from areas in the current configuration to areas in the reference configuration and vice versa.

This formula states that


where   is an area of a region in the current configuration,   is the same area in the reference configuration, and   is the outward normal to the area element in the current configuration while   is the outward normal in the reference configuration.


To see how this formula is derived, we start with the oriented area elements in the reference and current configurations:


The reference and current volumes of an element are


where  .







So we get